Option 2

where e is the membrane porosity, dp is the pore diameter, z is the viscosity of the fluid permeating the membrane, and 5m is the thickness of the membrane.

1.2.2.2.2 Mass Transfer Model

The gel polarization model has been successfully used to describe permeate flux in the pressure independent, mass transfer controlled region [52]. In this model, the solute is brought to membrane surface by convective transport. The resulting concentration gradient causes the solute to be transported back into bulk solution via diffusion. At steady state, these two mechanisms will balance each other and flux (J) can be integrated over the boundary layer to give [52]

Sg Cb

where D is diffusion coefficient, Sg is the thickness of gel layer, Cg is the gel concentration, and Cb is the bulk concentration. The diffusion coefficient, bulk, and gel layer concentrations are determined by physicochemical properties of feed. Equation 1.34 indicates that in the pressure independent region, flux can be improved through reducing thickness of gel layer by increasing shear rate, increasing diffusion coefficient by increasing temperature, or reducing bulk protein concentration.

Although the gel polarization model has been widely used in protein UF, this model was found to underestimate the flux of cross-flow MF [53]. The calculated filtrate flux based on this model is one to two orders of magnitude lower than experimental observations. The unexpected flux behavior observed in particle MF was referred to as the flux paradox.

where pL and m are fluid density and viscosity, yw is the wall shear rate, and rp is the particle radius.

Zydney and Colton [54] modified the gel-polarization model by replacing Brownian diffusivity with shear-induced diffusivity and proposed that the filtrate flux during cross-flow MF in an open channel could be described as